Cast turbine blades often include a structure known as a tip cap at the far end thereof. A tip cap is a relatively small extension of the blade. The tip cap may have a cross-sectional shape conforming to that of the blade and may be integral with or mounted on the radially outer end of the blade. After a period of time in service, the tip cap may be eroded, oxidized, and/or corroded by impingement of the hot combustion gases. Because the turbine blade may be expensive to produce, repair or refurbishment of the damaged blade may be preferred if possible. The repair of a turbine blade tip cap may be performed by grinding away the damaged material and then welding or otherwise attaching replacement material. Cooling the tip cap also may extend the overall useful component lifetime. Providing cooling, however, has been difficult due to manufacturing constraints.